Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions

Let $X$ be an $M\times N$ random matrices consisting of independent $M$-variate elliptically distributed column vectors $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ with general population covariance matrix $\Sigma$. In the literature, the quantity $XX^{*}$ is referred to as the sample covariance matrix, where $X^{*}$ is the transpose of $X$. In this article, we show that the limiting behavior of the scaled largest eigenvalue of $XX^{*}$ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit as $M,N\to\infty$ with $M/N\to\phi>0$ regardless of the distributions that $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ follow. In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled largest eigenvalue is the celebrated Tracy-Widom law. Applications of our results to the statistical signal detection problems have also been discussed.